Now we use the idea of recursion method. As shown in figure REF , suppose at the beginning we have found a bit thread configuration \(V\)  that can lock three regions \(\lbrace  A,B,AB\rbrace \)  simultaneously, or equivalently, lock \(\lbrace  {\gamma _A},{\gamma _B},{\gamma _{AB}}\rbrace  \) . In addition, we label the three-sided cell enclosed by \({\gamma _A}\) , \({\gamma _B}\) , \({\gamma _{AB}}\)  as \(\mathcal {C}\left( {AB} \right)\) , and the three two-sided cells as \(\mathcal {W}\left( A \right)\) , \(\mathcal {W}\left( B \right)\) , and \(\mathcal {W}\left( {\overline{AB} } \right)\)  (because they are entanglement wedges of \(A\) , \(B\) , and \(\overline{AB}\) , respectively). It can be observed that such a locking bit thread configuration has three types of thread bundles traveling in different directions in each three-sided cell (painted in white in the figure), and each type connects a pair of RT surfaces, while in each two-sided cell (painted in light gray), the threads simply travel in one direction, connecting the boundary subregion to the corresponding RT surface. Next, we can further decompose the two-sided cell surrounded by \({\gamma _{AB}}\)  and \(\overline{AB} \)  into a three-sided cell and two two-sided cells, and we are considering a corresponding problem: can we find a (new) bit thread configuration that can further lock the set of boundary subregions \(\lbrace  A,B,AB,C,ABC\rbrace  \)  simultaneously (or equivalently, lock \(\lbrace  {\gamma _A},{\gamma _B},{\gamma _{AB}},{\gamma _C},{\gamma _{ABC}}\rbrace  \) ). The idea is that we can replace only the part of the bit threads inside \(\mathcal {W}\left( {\overline{AB} } \right)\)  based on the original locking bit thread configuration \(V\)  [1]}, [2]}. This process can be referred to as the fine-graining of the bit thread configuration, since it can be accordingly understood as a further fine-graining of the entanglement structure of the boundary regions. One can see that by such a refinement process, the bit threads in \(\mathcal {W}\left( {\overline{AB} } \right)\)  that originally only extended in one direction were replaced by a thread configuration that have different types of thread bundles along different directions and might lead to intersecting threads (due to the requirement of “CFF=PEE”). In other words, more entanglement information was presented. Conversely, the original partial thread configurations in two-sided cells present merely coarse-graining information. Therefore, one can ask, can we always find such a more refined partial thread configuration (to replace the original one) satisfying the constraint of bit threads and “PEE=CFF” condition? Next we will use an explicit construction to show that the answer is positive. Once the proof is complete, as shown in figure REF , we can continue to iterate this refinement process, repeatedly subdividing the two-sided cells (gray areas in the figure) into three-sided cells (white areas) and smaller two-sided cells. Finally, we can obtain an exquisite locking bit thread configuration that can lock a large \(N\)  number of elementary regions and a set of non-crossing composite regions simultaneously while satisfy the “CFF=PEE” constraint at the same time.
